Design and Analysis of the Progressive Second

Price Auction for Network Bandwidth Sharing�

Aurel A. Lazar

Dept. of Electrical Engineering, Columbia University

New York, NY, 10027-6699, USA

aurel@comet.columbia.edu

and

Nemo Semrety

Dept. of Electrical Engineering, University of California, Los Angeles

Los Angeles, CA, 90095-1594, USA

nemo@ee.ucla.edu

April 1, 1998

Revised: September 30, 1999

Abstract

We present the Progressive Second Price auction, a new decen-tralized mechanism for allocation of variable-size shares of a resourceamong multiple users. Unlike most mechanisms in the economics lit-terature, PSP is designed with a very small message space, makingit suitable for real-time market pricing of communication bandwidth.Under elastic demand, the PSP auction is incentive compatible andstable, in that it has a \truthful" �-Nash equilibrium where all playersbid at prices equal to their marginal valuation of the resource. PSPis economically e�cient in that the equilibrium allocation maximizestotal user value. With simulations using a protype implementationof the auction game on the Internet, we investigate how convergencetimes scale with the number of bidders, as well as the trade-o� betweenengineering and economic e�ciency. We also provide a rate-distortion

�Parts of this work were presented at the 8th International Symposium on Dynamic

Games and Applications, Maastricht, The Netherlands, July 1998, and at the DIMACS

Workshop on Economics, Game Theory, and the Internet, Rutgers, NJ, April 1997.yCorresponding author.

1

theoretic basis for valuation of bandwidth, which leads naturally to theelastic demand model that is assumed in the analysis of the mechanism.

Keywords: resource allocation, auctions, game theory, mechanismdesign, network pricing.

1 Introduction

Communication networks are characterized by what economists call exter-nalities. The value a user gets from the network depends on the otherusers. The positive externalities are that a communication network is morevaluable if more people are connected. The negative externalities are thatresources are shared by users who { because of distance, population size, orsel�shness { cannot or will not coordinate their actions su�ciently to achievethe most desirable allocation of resources. The recognition of this reality inmany aspects of networks and distributed computations has lead in recentyears to the emergence of game theoretic approaches in their analysis anddesign [23, 9, 25, 33, 15, 16].

Prices, whether they relate to \real money" in a public network or \funnymoney" (based on quotas) in a private system, play a key role as allocationcontrol signals. In the former case, this role is of course intimately tied toanother, which is to allow a network provider to remain in business [7].

The telephone system and the current Internet represent two extremesof the relationship between resource allocation and pricing. The resourcesallocated to a telephone call are �xed, and usage prices are based on thepredictability of the total demand at any given time. On the Internet, thecurrent practice of pricing by the maximum capacity of the user's connection( at-rate pricing) decouples the allocation (actual use) of resources from theprices.

In the emerging multiservice networks (ATM, Next-Generation Inter-net), neither of these approaches are viable. The former because of the wideand rapidly evolving range of applications (including some which adapt toresource availability) will make demand more di�cult to predict. And thelatter because, once the at fee is paid, there are no incentives to limit usagesince increasing consumption bene�ts the user individually, whereas limitingit to sustainable levels brings bene�ts which are shared by all. This makesit vulnerable to the well-known \tragedy of the commons". With at pric-ing alone, the tendency is toward increasing congestion which chases awayhigh-value users, or increasing prices which exclude low-value users [6], inboth cases leading to decreased network revenue.

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Thus there is a need to develop new approaches to pricing of networkresources. Among the requirements are: sensitivity to the range of resourcerequirements (either through a su�ciently rich range of tra�c classes whichare priced di�erently, or by allowing users to explicitly quantify resourcerequirements); prices must be dynamically responsive to unpredictable de-mand (market based system); perhaps most importantly the pricing archi-tecture should constrain as little as possible the e�ciency trade-o�s of thepolicies.

Indeed, the fundamental issue in designing pricing policies is the trade-o� between engineering e�ciency and economic e�ciency. This trade-o�,which is more or less constrained by the underlying network technology, hasmany dimensions, including:

� how much measurement (from usage to capacity pricing),

� the granularity of di�erently priced service o�erings (e.g. number oftra�c classes),

� the level of resource aggregation { both in time and in space { at whichpricing is done (per packet/cell or per connection, at the edge of thenetwork or at each hop), and

� the information requirement (how much a priori knowledge of userbehavior and preferences is required/assumed by the network in com-puting prices).

An approach which achieves economic e�ciency is the smart-market ap-proach of [19], wherein each packet contains a bid, and if it is served, paysa clearing price given by the highest bid among packets which are deniedservice (dropped). This approach is incentive compatible in that the optimalstrategy for a (sel�sh) user is to set the bid price in each packet equal tothe true valuation. Each node in the network becomes an e�cient market,but the engineering cost (sorting packets by bid price, as well as per-packetand per-hop accounting) could be signi�cant if line speeds are high relativeto the processing power in the router. In [14], users are charged accordingto a combination of declared and measured characteristics of tra�c. Bytaking an equivalent bandwidth model of resource utilization, and assumingappropriate tra�c models, a menu of pricing plans indexed by the declaredtra�c can be o�ered which encourages users to make truthful declarations(e.g. of the mean rate), and also encourages the users' characterization ef-forts to be directed where they are most relevant to the network resource

3

allocation. As the pricing is relative, [14] does not aim to address the prob-lem of determining the actual monetary values of the market price (thatusers would be willing to pay). Another pricing scheme which incorpo-rates multiplexing gain is formulated in [12]. These and a number of otherschemes are summarized in [11], in a comprehensive view of the connectionestablishment process, which identi�es the user-network negotiation as thekey \missing link" in network engineering/economic research. In terms ofour taxonomy of the previous paragraph, this is part of the information re-quirement trade-o�. Indeed, in the absence of formal mechanisms to dealwith the information problem, complex and (at least intuitively) undesirablethings happen. For example, some providers o�er expensive \front of thebook" rates to uninformed customers, and lower \back of the book" rates toinformed customers who may be about to defect to another carrier (see [7]and also the recent wars between AT&T and MCI in consumer long-distanceservice in the United States). In [34], it is argued that architectural consid-erations such as where charges are assessed should take precedence over thepursuit of optimal e�ciency, and edge pricing (spatial aggregation in termsof our taxonomy) is proposed as a useful paradigm.

In this paper, we propose a new auction mechanism which accommodatesvarious dimensions of the engineering-economics trade-o�. The mechanismapplies to a generic arbitrarily divisible and additive resource model (whichmay be equivalent bandwidth, peak rate, contract regions, etc., at any levelof aggregation.) It does not assume any speci�c mapping of resource allo-cation to quality of service. Rather, users are de�ned as having an explicitmonetary valuation of quantities of resource, which the network doesn't orcan't know a priori. Thus, in terms of our trade-o� taxonomy, this mech-anism aims for unlimited granularity, exibility in the level of aggregationand minimal information requirement.

In the most likely auction scenaria, users would be aggregates of many ows data ows for which bulk capacity is being purchased for e.g. VirtualPaths, Virtual Private Networks, or edge capacity [2, 31].

We begin in Section 2 by formally presenting the design of our Progres-sive Second Price auction mechanism for sharing a single arbitrarily divisibleresource, and relating it to classical mechanism design from the economicslitterature. In Section 3, after describing our model of user preferencesand the elastic demand assumption, we prove that PSP has the desiredproperties of incentive compatibility, stability, and e�ciency. The sectionconcludes with simulation results on the convergence properties, and thee�ciency trade-o�s. Appendix A describes an information theoretic basis

4

for valuations of the type that are assumed in the analysis of Sections 3, asone possible justi�cation.

2 Design of an Auction for a Divisible Resource

2.1 Message Process

Following [36], it is useful to expose the design in terms of its two aspects:realization, where a message process that enables a certain allocation ob-jective is de�ned; and Nash implementation, where allocation rules are de-signed with incentives which drive the players to an equilibrium where the(designer's) desired allocation is achieved.

In this section we de�ne the message process. Here we make the fun-damental choice which will constrain the subsequent aspects of the design.Our �rst concern here is with engineering. For the sake of scalability in anetwork setting, we shall aim for a process where a) the exchanged messagesare as small as possible, while still conveying enough information to allowresource allocation and pricing to be performed without any a-priori knowl-edge of demand (market research, etc.); and b) the amount of computationat the center is minimized.

Given a quantity Q of a resource, and a set of players I = f1; : : : ; Ig,an auction is a mechanism consisting of: 1) players submitting bids, i.e.declaring their desired share of the total resource and a price they are willingto pay for it, and 2) the auctioneer allocating shares of the resource to theplayers based on their bids.

Player i's bid is si = (qi; pi) 2 Si = [0; Q]� [0;1), meaning he would likea quantity qi at a unit price pi. A bid pro�le is s = (s1; : : : ; sI). Followingstandard game theoretic notation, let s�i � (s1; : : : ; si�1; si+1; : : : ; sI), i.e.the bid pro�le of player i's opponents, obtained from s by deleting si. Whenwe wish to emphasize a dependence on a particular player's bid si, we willwrite the pro�le s as (si; s�i).

The allocation is done by an allocation rule A,

A : S �! Ss = (q; p) 7�! A(s) = (a(s); c(s));

where S =Qi2I Si.

The i-th row of A(s), Ai(s) = (ai(s); ci(s)), is the allocation to playeri: she gets a quantity ai(s) for which she is charged ci(s). Note that p is aprice per unit and c is a total cost.

5

An allocation rule A is feasible if 8s,Xi2I

ai(s) � Q

and 8i 2 I,ai(s) � qi;

ci(s) � piqi:

Remark a: The above formulation is a generalization of what is usuallymeant by an auction. The latter is the special case where aw(s) = Q forsome winner w 2 I and ai(s) = 0; 8i 6= w, i.e. the sale of a single indivisibleobject to one buyer, for which the theory is well developed [22, 24]. In ourapproach, allocations are for arbitrary shares of the total available quantityof resource. Equivalently, one could slice the resource into many small units,each of which is auctioned as an indivisible object. But in a practical imple-mentation of auctions for sharing a resource, a process of bidding for eachindividual unit would result in a tremendous signaling overhead. More im-portantly, since the users would be bidding on a discrete grid of quantities,analytical predictions of outcomes could be misleading since they could besensitive to the particular choice of grid1.

Remark b: Most of the mechanism design literature in Economicsmakes use of the following \Revelation Principle":

Given any feasible auction mechanism, there exists an equivalent2

feasible direct revelation mechanism which gives to the seller andall bidders the same expected utilities as the given mechanism.([24], Lemma 1)

In this sharing context, a direct revelation mechanism would be one whereeach user message consists of the user's type, which is the valuation3 of theresource over the whole range of their possible demands, i.e. a function�i : [0; Q] ! [0;1), and the budget (see Section 3.1). A consequence ofrevelation principle is that the mechanism designer can restrict her atten-tion to direct revelation mechanisms, �nd the best mechanism in terms of

1For a more detailed discussion of this point, see [4] p. 34, and references therein.2By equivalent, in [24] it is meant that, at some equilibrium, all players get the same

utility. There may be other, possibly ill-behaved, equilibria.3The valuation of a given amount of resource is how much the user is willing to pay for

that quantity. The inverse of the valuation is the user's demand function, giving a desiredquantity for each price.

6

her (economic) e�ciency objectives, and then { if necessary { transform itinto an equivalent mechanism in the desired message space. This is conve-nient because one can exclude the in�nitely many mechanisms with largermessage spaces, without fear of missing any better designs. The design pro-cess is usually the solution of an optimization (mathematical programming)problem. For this reason, in the litterature, mechanism design problems aremostly solved for cases where the space of users' types is one dimensional,or at most �nite dimensional [21], using message spaces that are of the samedimension.

In our sharing problem, the conventional approach is unsatisfactory intwo ways:

� First, a user's type is in�nite-dimensional, as we do not restrict thevaluation functions beyond some very general assumptions (see Sec-tion 3.1), and so the conventional \programming" approach of de-riving the mechanism from the revelation principle would lead to anintractable problem.

� Second, the conventional (direct revelation) approach, even if it wastractable, implies that a single message (bid) can theoretically be in-�nitely long, because it has to contain a description of the function �i.Clearly, this is not desirable in a communication network, where sig-naling load is a key consideration. For engineering reasons, we choosea message space that is 2-dimensional. Therefore, a given message cancome from many possible types, so there is no single way to do thetransformation from the direct revelation mechanism to the desiredone.

Thus, unlike most of the mechanism design literature, we will take a di-rect approach, where we posit an allocation rule for our desired messagespace, and then show that it has an equilibrium, and that the design objec-tive is met at equilibrium4 . This is equivalent to guessing the right direct-revelation-to-desired-mechanism transformation and building it into the al-location rule from the start.

4Our aim is to to show that we can use this smaller message space and still achieve ourobjective.

7

Q

p i

q 2

p 0

q 1

q i

q

q 0

q 4

5p 5

p 4

p 2

s i

p 1

ai = Q - q5 - q4

ci =

Figure 1: Exclusion-compensation principle: the intuition behind the PSPrule

2.2 Allocation Rule

De�ne, for y � 0

Qi(y; s�i) =

24Q�

Xpk�y;k 6=i

qk

35+

: (1)

and

Qi(y; s�i) = lim�&y

Qi(�; s�i) =

24Q�

Xpk>y;k 6=i

qk

35+

:

The \progressive second price" (PSP) allocation rule is de�ned as follows:

ai(s) = qi ^ Qi(pi; s�i); (2)

ci(s) =Xj 6=i

pj [aj(0; s�i)� aj(si; s�i)] ; (3)

where ^ means taking the minimum.Remark a: For a �xed opponent pro�le s�i, Qi(pi; s�i) represents the

maximum available quantity at a bid price of pi. The intuition behind

PSP is an exclusion-compensation principle: player i pays for his allocation

8

so as to exactly cover the \social opportunity cost" which is given by the de-clared willingness to pay (bids) of the users who are excluded by i's presence(see Figure 1), and thus also compensates the seller for the maximum lostpotential revenue. Note that this amounts to implicitly assuming that thebid price accurately re ects the marginal valuation �0i on the range [ai; qi].In other words, by this rule the auctioneer is saying to the player: \if youbid (qi; pi), I take it to mean that in the vicinity of qi, �i can be approxi-mated by a line of slope pi." This is the (built-in) transformation from thedirect-revelation mechanism to the desired message process discussed in thesecond remark at the end of Section 2.1.

The charge ci increases with ai in a manner similar to the income tax ina progressive tax system. For a �xed opponent pro�le s�i, imagine player iis increasing qi, starting from 0. The �rst few units that player i gets will betaken away from the lowest clearing opponent (i.e. m = argminjfpj : aj >0g), and player i will pay a price (marginal cost) pm per unit. When amreaches 0, the subsequent units that player i gets will cost him pm0 > pm,where m0 is the new lowest clearing player, the one just above m. ThePSP rule is the natural generalization of second-price auctions (or Vickreyauctions). In a Vickrey auction of a single non-divisible object, each playersubmits a sealed bid, and the object is sold to the highest bidder at the bidprice of the second highest bidder, which is what happens here if qi = Q; 8i.This is widely known to have many desirable properties [35, 24, 4], the mostimportant of which is that it has an equilibrium pro�le where all players bidtheir true valuation. As we will presently show, this property is preservedby the PSP rule in the more general case of sharing an arbitrarily divisibleresource, and this leads to stability (Nash equilibrium). The PSP rule isanalogous to Clarke-Groves mechanisms [3, 8, 20] in the direct-revelationcase.

Remark b: When two players bid at exactly the same price, and theyare asking for more than is available at that price, (2) punishes both of them.For example, if Q = 100 and s1 = (4; 60) and s2 = (4; 70), the allocationswould be a1 = 60 ^ (100� 70) = 30, and a2 = 70 ^ (100� 60) = 40. Sincethe bid prices are equal, there is no \right" way to decide who to give theremaining capacity to. One could divide it equally, or proportionally to theirrequests, etc. For the subsequent analysis, it turns out it is simpler to notgive it to either one (of course, it will be allocated to the lower bidders ifthere are any). This is just a technicality since by deciding this, we ensurethat it will never happen (at equilibrium), since the users will always preferto change their prices and/or reduce their quantity.

9

Considering the computational complexity of PSP, a straightforward im-plementation would at worst, sort the bids in time I log I , perform (2) inlinear time, and (3) can be done in time I2. Thus, the complexity of com-puting the allocations is O(I2).

3 Analysis of the Progressive Second Price Auc-

tion

3.1 User Preferences

Since the allocation rule A is given by design, the only analytical assump-tions we make is on the form of the players' preferences.

Player i's preferences are given by his utility function

ui : S �! (�1;1)s 7�! ui(s):

Player i has a valuation of the resource �i(ai(s)) � 0, which is the totalvalue to her of her allocation. Thus, for a bid pro�le of s, under allocationrule A, player i getting an allocation Ai(s) has the quasi-linear utility

ui(s) = �i(ai(s))� ci(s) (4)

which is simply the value of what she gets minus the cost.In addition, the player can be constrained by a budget bi 2 [0;1], so

the bid si must lie in the set

Si(s�i) = fsi 2 Si : ci(si; s�i) � big: (5)

In the proofs of the following section, we will assume that users haveelastic demand, that is:

Assumption 1 For any i 2 I,� �i(0) = 0;

� �i is di�erentiable,

� �0i � 0, non-increasing and continuous

� 9 i > 0, 8z � 0, �0i(z) > 0) 8� < z; �0i(z) � �0i(�)� i(z � �).

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The last item says that as long as the valuation is strictly increasing, it mustalso be strictly concave (with minimum curvature i). However, it is allowedto \ atten" beyond a certain amount of resource.

Functions of this (concave) form have wide applicability as models ofresource valuation, and can be justi�ed from the economic standpoint (di-minishing returns) as well as from information theoretic standpoint { seeAppendix A. For examples of valuations satisfying Assumption 1, see Sec-tion 3.4 and Appendix A.

3.2 Equilibrium of PSP

The auction game is given by (Q; u1; : : : ; uI ; A), that is, by specifying theresource, the players, and a feasible allocation rule. We analyze it as astrategic game of complete information [4].

De�ne the set of best replies to a pro�le s�i of opponents bids: S�i (s�i) =fsi 2 Si(s�i) : ui(si; s�i) � ui(s

0i; s�i); 8s0i 2 Si(s�i)g. Let S�(s) = Q

i S�i (s�i).

A Nash equilibrium is a �xed point of the point-to-set mapping S�, i.e. apro�le s 2 S�(s). In other words, it is a point from which no player willwant to unilaterally deviate. Such a point is what is most accepted as aconsistent prediction of the actual outcome of a game, and has been re-peatedly con�rmed by experiments, as well as a wide range of theoreticalapproaches. Indeed, in a dynamic game, where players recompute the bestresponse to the current strategy pro�le of their opponents, this iterationcan only converge to a Nash equilibrium (if it converges at all). In addition,an important trend in modern game theory is the development of learningmodels, and there too, it has been shown that Nash equilibria result alsofrom rational learning through repeated play among the same players [13].

A more general (and hence weaker) notion of stability is the existenceof an �-Nash equilibrium. Let the �-best replies be S�i (s�i) = fsi 2 Si(s�i) :ui(si; s�i) � ui(s0i; s�i)� �; 8s0i 2 Si(s�i)g. An �-Nash equilibrium is a �xedpoint of S�.

In a dynamic auction game, � > 0 can be interpreted as a bid fee paidby a bidder each time they submit a bid. Thus, the user will send a bestreply bid as long as it improves her current utility by �, and the game canonly end at an �-Nash equilibrium.

De�nePi(z; s�i) = inf fy � 0 : Qi(y; s�i) � zg : (6)

11

Thus, for �xed s�i, 8y; z � 0,

z � Qi(y; s�i)) y � Pi(z; s�i) (7)

and5

y > Pi(z; s�i)) z � Qi(y; s�i): (8)

The graph of Pi(:; s�i) is the \staircase" shown in Figure 1, and that ofQi(:; s�i) is obtained by ipping it 90 degrees.

It is readily apparent that

ci(s) =Z ai(s)

0Pi(z; s�i) dz: (9)

The key property of PSP is that, for a given opponent pro�le, a playercannot do much better than simply tell the truth, which in this setting meansbidding at a price equal to the marginal valuation, i.e. set pi = �0i(qi). Bydoing so, she can always get within � > 0 of the best utility.

Let Ti = fsi 2 Si : pi = �0i(qi)g, the (unconstrained) set of player i'struthful bids, and T =

Qi Ti.

Proposition 1 (Incentive compatibility) Under Assumption 1, 8i 2 I,8s�i 2 S�i, such that Qi(0; s�i) = 0, for any � > 0, there exists a truthful�-best reply ti(s�i) 2 Ti \ S�i (s�i).

In particular, let

Gi(s�i) =

�z 2 [0; Q] : z � Qi(�

0i(z); s�i) and

Z z

0Pi(�; s�i) d� � bi

�:

Then with vi = [supGi(s�i)� �=�0i(0)]+ and wi = �0i(vi), ti = (vi; wi) 2

Ti \ S�i (s�i).

The truthful best reply can be found in a straightforward manner, asillustrated in Figure 2.Proof: Fix s�i 2 S�i. Let zi = supGi(s�i) and yi = �0i(zi).

By de�nition of zi, 9fz(n)g � Gi(s�i) such that limn z(n) = zi. Hence bi �limn

R z(n)0 Pi(�; s�i) d� =

R zi0 Pi(�; s�i) d� � ci(ti; s�i), where the equality comes

from the boundedness of Pi and the Lebesgue dominated convergence theorem, andthe second inequality from (9) and (2). Thus ti 2 T \ Si(s�i).

5Actually, since Qi(:; s�i) is upper-semi-continuous (jumps up), we have z �

Qi(y; s�i), y � Pi(z; s�i)

12

Q

iθ’

t i

ci =

ui =

Figure 2: Truthful �-best reply

Next we show that ti 2 S�i (s�i). First, zi = limn z(n) � limnQi(�0i(z(n)); s�i) �Qi(limn �

0i(z(n)); s�i), where the inequalities follow respectively from z(n) 2 Gi(s�i),

and the upper semi-continuity ofQi(:; s�i). Now by the continuity of �0i, Qi(limn �0i(z(n)); s�i) =

Qi(�0i(zi); s�i) = Qi(yi; s�i), hence

zi � Qi(yi; s�i): (10)

Now, we claim that ai(ti; s�i) = vi. Indeed, if zi = 0 then vi = 0 and ai(ti; s�i) = 0.If zi > 0, then by (10), Qi(yi; s�i) > 0 and since by hypothesis Qi(0; s�i) = 0, wehave �0i(zi) = yi > 0. Also, zi > 0 implies vi < zi. Therefore, by Assumption 1,we have wi = �0i(vi) > �0i(zi) = yi. Hence, since Q

i(:; s�i) is non-decreasing,

Qi(wi; s�i) � lim�&y Qi

(�; s�i) = Qi(yi; s�i) � zi > vi. Thus, by (2),

ai(ti; s�i) = vi (11)

Now 8si 2 Si(s�i),

ui(ti; s�i) � ui(s)

= �i(ai(ti; s�i)) � �i(ai(s)) � ci(ti; s�i) + ci(s)

=

Z ai(s)

ai(ti;s�i)[Pi(z; s�i)� �0i(z)] dz:

=

Z ai(s)

zi

[Pi(z; s�i)� �0i(z)] dz +Z zi

vi

[Pi(z; s�i)� �0i(z)] dz

13

�Z ai(s)

zi

[Pi(z; s�i)� �0i(z)] dz � � (12)

where the inequality follows from (zi � vi) � �=�0i(0) and the fact that �0i is non-increasing. Thus, it su�ces to show that the integral is � 0.

If zi < ai(s), take any z 2 (zi; ai(s)]. By the de�nition of zi, z 62 Gi(s�i). Nowsi 2 Si(s�i) implies bi � ci(s) =

R ai(s)0 Pi(�; s�i) d� �

R z0 Pi(�; s�i) d�. Therefore,

we must have z > Qi(�0i(z)), which by (8), implies �0i(z) � Pi(z) and the integrandin (12) is � 0 as desired.

Suppose zi � ai(si). Since �0i is non-increasing, Qi(:; s�i) is non-decreasing and

Pi(:; s�i) � 0, any point to the left of zi is in the set Gi(s�i), 8z < zi; z 2 Gi(s�i),hence z � Qi(�0i(z); s�i) which by (7), implies �0i(z) � Pi(z; s�i), so the integrand

in (12) is � 0 as desired. 2

Figure 3 shows the utility function of player 4, u4(s4), in a PSP auctionwith I = 5 players, with s�4 �xed, and a valuation �4(q) = 10q. The plateaus

correspond to the points where q4 � Q4(p4; s) =hQ�P

fj:pj>p4g aj(s)i+,

and a4(s) can no longer be increased at that bid price { see (2). At bid pricesp4 > p5, the utility decreases when a4 > Q � q5, because after that point,each additional unit of resource is taken away from player 5, and thus costsp5, which is more than �04 its value to player i. Thus, each additional unitstarts bringing negative utility. This is what discourages users from biddingabove their valuation. Proposition 1 is illustrated by the fact that for anygiven quantity q4, the utility u4 is maximized on the plane p4 = �04 � 10.

Remark: When the players have linear valuations and no budget con-straint (bi = 1), PSP becomes identical to a second-price auction for anon-divisible object. Then the existence of a Nash equilibrium follows di-rectly from incentive compatibility.

Note that in PSP, the incentive compatibility (optimality of truth-telling)is in the price dimension, for a given quantity. With the message space wehave designed, there is no single \true" quantity to declare, the optimalquantity depends on opponent bid prices. Were the message process suchthat players declared a price and a budget (rather than desired quantity),it may have been possible to design an allocation rule A such that they areinclined to reveal their true budget, thus obtaining incentive compatibility inboth dimensions, and hence equilibrium. But such a rule A would likely nothave a simple closed form like (2) and (3). In essence, the computational loadof translating budgets into shares would be centralized at the auctioneer,thus making the system less scalable to large numbers of users. On the

14

0

20

40

60

80

100

0

5

10

150

100

200

300

400

500

quantity −− q4price −− p4

util

ity −

− u

4

Figure 3: Utility u4(s4) for s1 = (100; 1), s2 = (10; 2), s3 = (20; 4), s5 =(20; 7), s6 = (30; 12)

other hand, decentralization has a cost too, which is the signaling overheadresulting from players possibly adjusting bids based on opponent bids in theiterated game. Our design is based on the premise that the latter approachis the more scalable of the two (indeed that was the reason for choosing asmall message space).

The next property is that the truthful best reply is continuous in op-ponent pro�les (this can be seen in Figure 2: as the \staircase" is variedsmoothly, the point of intersection with �0i moves smoothly, provided �0i isnot at { which is given by the last time item in Assumption 1). To provethat, we will need the following:

Lemma 1 8s; s0 2 S; 8y; z � 0; 8� > 0, if jjs�i � s0�ijj < � then

Qi(y + �; s�i) + �pI � Qi(y; s

0�i) � Qi(y � �; s�i)� �

pI; (13)

andPi(z + �

pI; s�i) + � � Pi(z; s

0�i) � Pi(z � �

pI; s�i)� �: (14)

Proof: First, jjs�i�s0�ijj < � impliesP

k jq0k�qkj < �pI , and pk+� > p0k > pk��.

Thus,P

k qk1fpk+�>yg + �pI � P

k q0k1fp0k>yg �

Pk qk1fpk��>yg � �

pI: Then,

using (1) and the identity (a + b)+ � (a)+ + (b)+, the �rst result follows.

15

For any y < Pi(z; s0�i), by (7), we have z > Qi(y; s0�i) � Qi(y � �; s�i) � �pI,

which by (8), implies y � � � Pi(z + �pI; s�i). Letting y % Pi(z; s0�i), we get

Pi(z; s0�i) � Pi(z + �

pI; s�i) + �.

For any y > Pi(z; s0�i), by (8), we have z � Qi(y; s

0�i) � Qi(y + �; s�i) + �

pI,

which by (7) implies y + � � Pi(z � �pI). Letting y & Pi(z; s0�i), we get

Pi(z; s0�i) � Pi(z � �pI)� �. 2

Lemma 2 (Continuity of best reply) Under Assumption 1, 8i 2 I, the �-bestreply ti given in Proposition 1 is continuous in s�i on any subset Vi(P; P ) =fs�i 2 Si : 8z > 0; P � Pi(z; s�i) � P g, with 1 > P � P > 0.

Proof: Let zi = supGi(s�i). We will show zi is continuous, and the continuity ofvi = [zi��=�0i(0)]+ and wi = �0i(vi) follow immediately (recall that by Assumption 1,�0i is continuous).

Suppose there is a discontinuity at some s�i. Then, 9�1 > 0, such that 8� > 0,9s0�i 2 Vi(P ; P ) with jjs�i � s0�ijj < � and jjzi � z0ijj � �1, where z0i = supGi(s0�i).

Suppose zi+ �1 � z0i = supGi(s0�i) (the case z0i+ �1 � zi is handled identically,with s�i and s0�i interchanged. ). Consider the de�nition of Gi; since �0i is decreas-ing and Qi(:; s�i) is non-decreasing and Pi(:; s�i) � 0, any point to the left of z0i isin the set Gi(s0�i), therefore

zi + �1 2 Gi(s0�i): (15)

Thus, zi + �1 � Qi(�0i(zi + �1); s0�i) =hQ�Pk q

0k1fp0k>�0i(zi+�1)g

i+. Therefore,

zi + �1 � Qi(�0i(zi + �1) + �; s�i) + �

pI;

using Lemma 1.Also, by (7) zi + �1 � Qi(�0i(zi + �1); s0�i)) �0i(zi + �1) > P (zi + �1; s

0�i). Now

since s0�i 2 Vi(P ; P), this last expression is � P > 0, hence �0i(zi + �1) > 0. Thenusing Assumption 1, �0i(zi + �2) � �0i(zi + �1) + i(�1 � �2) > �0i(zi + �1) + �, for� < �1 = �1p

I^ �1 i, and 0 < �2 < (�1 � �1

pI) ^ (�1 � �1= i). Therefore, since

Qi(:; s�i) is non-decreasing,

zi + �2 � Qi(�0i(zi + �2); s�i) + �

pI � �1 + �2

< Qi(�0i(zi + �2); s�i): (16)

Now (15) also implies that

bi �Z zi+�1

0

Pi(�; s0�i) d�

�Z zi+�3

0

Pi(�; s�i) d� +Z zi+�3

0

�Pi(�; s

0�i) � Pi(�; s�i)

�d� + (�1 � �3)P ;

16

and this holds 8�3 < �1. Now, using Lemma 1,

Z zi+�3

0

�Pi(�; s

0�i)� Pi(�; s�i)

�d�

� ��Q+

Z zi+�3

0

Pi(� � �pI; s�i) d� �

Z zi+�3

0

Pi(�; s�i) d�

� ��Q�Z zi+�3

zi+�3��pI

Pi(�; s�i) d�

� �(Q + P )�pI:

Let �2 =�1P

(Q+P )pI, and �3 such that 0 < �3 < �1 � (Q + P )�2

pI=P . Then

bi �Z zi+�3

0

Pi(�; s�i) d�; (17)

for � < �2.

Now choosing � < �1^�2, (16) and (17) imply thatGi(s�i) 3 (zi+�3)^(zi+�2) >zi = supGi(s�i), a contradiction. 2

We introduce one additional player, player 0, whose valuation is �0(z) =p0z, and whose bid can therefore be �xed at s0 = (q0; p0) = (Q; p0). Player0 can be viewed as the auctioneer, and p0 > 0 as a \reserve price" at whichthe seller is willing to \buy" all of the resource from himself. From (1), thepresence of the bid s0 = (Q; p0) implies 8i 2 I; Qi(y; s�i) = 0; 8y < p0. Inparticular, setting y = 0, the condition of Proposition 1 holds. Thus, we canrestrict our attention to truthful strategies only, and still have feasible bestreplies. This forms a \truthful" game embedded within the larger auctiongame, where the strategy space is T � S, the feasible sets are Ti \ Si(s�i),and the best replies are R�

i(s) = Ti \ S�i (s). A �xed point of R� in T is a�xed point of S� in S. Thus an equilibrium of the embedded game is anequilibrium of the whole game.

Proposition 2 (Nash equilibrium) In the auction game with the PSP rulegiven by (2) and (3) and a reserve price p0 > 0, and players described by (4)and (5), if Assumption 1 holds, then for any � > 0, there exists a truthful�-Nash equilibrium s� 2 T .Proof: 8s 2 T , 8i 2 I, 8z > 0, we have z > 0 = Qi(p0=2; s�i), which by (8)

implies Pi(z; s�i) � p0=2 = P . Let P = maxk2I[f0g �0k(0). Then, the conditions

of Lemma 2 are satis�ed and t = (v; w) is continuous in s on T . By Assump-

tion 1, �0i is continuous therefore v(q; p) = v(q; �0(q)) (as de�ned in Proposition 1),

17

can be viewed as a continuous mapping of [0; Q]I onto itself. By Brouwer's �xed-

point theorem (see for example [10]), any continuous mapping of a convex compact

set into itself has at least one �xed point, i.e. 9q� = v(q�) 2 [0; Q]I. Now with

s� = (q�; �0(q�)), we have s� = t(s�) 2 T . 2

3.3 E�ciency

The objective in designing the auction is that, at equilbrium, resources al-ways go to those who value them most. Indeed, the PSP mechanism doeshave that property. This can be loosely argued as follows: for each player,the marginal valuation is never greater than the bid price of any opponentwho is getting a non-zero allocation. Thus, whenever there is a player jwhose marginal valuation is less than player i's and j is getting a non-zero allocation, i can take some away from j, paying a price less than i'smarginal valuation, i.e. increasing ui, but also increasing the total value,since i's marginal value is greater. Thus at equilibrium, i.e. when no onecan unilaterally increase their utility, the total value is maximized. Formally,consider a 2 argmaxA

Pi �i(ai). The Karush-Kuhn-Tucker [17] optimality

conditions are that there exists a Lagrange multiplier � such that �0i(ai) = �,if ai > 0, and �0i(0) � �, if ai = 0.

Assumption 2 For any i 2 I, bi =1, and �0i satis�es6

�0i(z)� �0i(z0) > ��(z � z0);

whenever z > z0 � 0.

Given any � > 0, for any �-Nash-equilbrium s� 2 T , let a� � a(s�), andlet a� � mini2I[f0g;ai>0 a

�i , the smallest non-zero allocation. The following

is the \� version" of the Karush-Kuhn-Tucker conditions.

Lemma 3 Suppose Assumptions 1 and 2 hold. If for some j, a�j >p�=�,

then 8i 2 I [ f0g,�0i(a

�i ) < �0j(a

�j ) + 2

p��:

An immediate corollary is that if a� >p�=� then

�� � 2p�� < �0i(a

�i ) < �� + 2

p��

6If �0i is di�erentiable, the condition is 0 � �00i > �� .

18

if a�i >p�=� and

�0i(a�i ) < �� + 2

p��;

if a�i = 0, for some �� � 0.

Proof: Suppose �0i(a�i ) � �0j(a

�j ) + 2

p��. Since �0j(a

�j ) � �0j(q

�j ) � p�j , we have

�0i(a�i ) � p�j + 2

p��:

Since a�j > 0, if player i bids at a price above p�j , he can take all of player j'sallocation, without losing anything of his own, i.e. a�i + a�j � Qi(p�j ; s

��i): By (7),

this impliesp�j � Pi(a

�i + a�j ; s

��i):

Let qi = (a�i +p�=�) and si = (qi; �0i(qi)). Then

ui(si; s��i)� ui(s

�) =

Z a�i+p

�=�

a�i

�0i(z) � Pi(z; s��i) dz

�h�0i(a

�i +

p�=�) � p�j

ip�=�

�h�0i(a

�i )� �

p�=�� p�j

ip�=�

�h2p��� �

p�=�

ip�=�

= �

which contradicts the fact that s� is an �-Nash equilibrium. 2

Proposition 3 (E�ciency) Suppose Assumptions 1 and 2 hold. If a� >p�=�, then

maxA

Xi

�i(ai)�Xi

�i(a�i ) = O(

p��);

where A = fa 2 [0; Q]I+1 :P

i ai � Qg.Proof: (of Proposition 3) Let I+ = fk : ak > a�kg and I� = fk : ak < a�kg. Fori 2 I+, we have �0i(a

�i ) � �� + 2

p��. For i 2 I�, we have a�i > ai � 0, therefore

by the lemma, �0i(a�i ) > �� � 2

p��. Therefore,

XI

�i(ai) � �i(a�i ) �

XI+

�0i(a�i )(ai � a�i ) �

XI�

�0i(a�i )(a

�i � ai)

� (�� + 2p��)�� (�� � 2

p��)�;

where � =P

I+ (ai � a�i ) =P

I�(a�i � ai). Since � � Q the result follows, with

the bound 4Qp��. 2

19

Remark a: The condition bi = 1 is su�cient, but not necessary toachieve e�cient outcomes. In fact with any budget pro�le, e�ciency canbe achieved if the users cooperate. For example, if they all choose a bidquantity close to what they can actually obtain (which they do if they usethe strategy given by Proposition 1), then the price paid would be p0 perunit for all the allocations, and if p0 or the shares a�i are not too large,then budget constraints are irrelevant and a� is e�cient. More generally,e�ciency is attained if the budgets are not too far out of line with thevaluations, i.e. there are no players with very high demand and very lowbudget.

Remark b: (Welfare and E�ciency) A more common measure of e�-ciency is the social welfare, which is the sum of all the players' utility

Pi ui,

including the seller i = 0. The natural de�nition of the seller's utility is thevalue of the leftover capacity a0 = Q�P

i6=0 ai plus the revenue, i.e.

u0 = �0(a0) +Xi6=0

ci:

Then,P

i ui =P

i6=0(�i � ci) + u0 =P

i6=0 �i + �0(a0) =P

i �i. Thus,P

i uiis equivalent to the e�ciency measure used above, which is

Pi �i. Another

measure is the seller's revenue. Even though PSP is not, in general, revenue-maximizing, it tends to the revenue maximizing allocations and prices asdemand increases [18].

Remark c: Proposition 3 provides a key to understanding the basictrade-o� between engineering and economic e�ciency. The smaller �, thecloser we get to the value-optimal allocations. But in a dynamic game,where players iteratively adjust their bids to the opponent pro�le, a playerwill bid as long as he can gain at least � utility (since that is the cost ofthe bid), thus a smaller � makes the iteration take longer to converge, i.e.entails more signaling.

3.4 Convergence

An issue of obvious concern is whether the game converges under dynamicplay: it turns out that it does, when users behave rationally (see Proposition4 in Chapter 2 of [29]). Moreover, irrational or malicious behavior { likeintentionally trying to prevent convergence by making unnecessary bids {can always be controlled by setting the bid fee � high enough to make suchbehavior prohibitively costly for the culprit.

20

0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

quantity

valu

atio

n

Figure 4: Parabolic valuation with � = 0:5 and q = 70

Another issue is how the convergence time scales with the number ofbidders. We now consider this experimentally using the software describedin Appendix B.

In all our simulations we let Q = 100. For each user, the valuation isstrictly increasing and concave up to a maximum corresponding to a physicalline capacity, and at beyond that. Since, as shown by Proposition 3, onlythe second derivative of the valuation is needed to measure the e�ciencyof the PSP auction, a second order (parabolic) model is deemed su�cient.Thus we use valuations of the form:

�i(z) = ��i(z ^ qi)2=2 + �iqi(z ^ qi);

where qi is the line rate, and �i > 0 (see Figure 4).We generate our user population with independent random variables

f�0i(0)gI (corresponding to the maximum unit price the user would pay)uniformly distributed on [10; 20], and �i = �0i(0)=qi, and qi uniformly dis-tributed on [50; 100]. All players have a budget bi = 100. The bid fee is�xed at � = 5. Each user has a bidding agent which can submit at most onebid per second (see Algorithm 1 in Appendix B).

With this set-up, the results are shown in Figures 5-6. Simulations wererun for 11 population sizes ranging from 2 to 96 players. Each point is

21

0 20 40 60 80 100 120 140 1600

500

1000

1500

2000

2500

3000

number of players

num

ber

of b

ids

to c

onve

rge

Figure 5: Mean (+/- std. dev.) number of bids { solid line. The dashedline is I + I2=10.

simulated 10 times with new random valuations for all players. The overallmean is 11.9 bids per player. From Figure 5, the number of bids seems togrow as the square of the number of players.

The actual time to converge, shown in Figure 6, grows more slowly, sincethe computation of bids is done in parallel by all the players. In fact, forsmall numbers of players, the time decreases. This can best be explained asfollows. Suppose there are only two players, with similar valuations. Theywill both start by asking for their maximum quantity, at their marginalvaluation (which at their maximum quantity is near zero). Then as eachsees the other's bid, each will reduce the quantity and increase the price alittle bit. And they go on taking turns, gradually raising the market priceuntil they reach an equilibrium. However if there are 10 players, in betweentwo bids by the same player, the 9 others will already have bid up theprice, so he will jump to higher price than if there was only one opponent.Thus the equilibrium market price will be reached more quickly. For largepopulations, this e�ect becomes small compared to the sheer volume of bids,and the convergence time starts to grow.

The trade-o� between signaling and economic e�ciency discussed in lightof Proposition 3 is illustrated by Figures 7-8. Increasing the bid fee speeds

22

0 20 40 60 80 100 120 140 1600

20

40

60

80

100

120

number of players

time

to c

onve

rge

(sec

onds

)

Figure 6: Mean (+/- std. dev.) convergence time in seconds (for a 1 secondbid interval).

up convergence, at a cost of lost e�ciency. A resource manager should selecta bid fee which optimally balances the two for the particular context.

Figure 8 also illustrates the validity of the lower bound given by Propo-sition 3.

4 Conclusion

Auctions are one of oldest surviving classes of economic insti-tutions [...] As impressive as the historical longevity is the re-markable range of situations in which they are currently used.[22]

We proposed the progressive second price auction, a new auction whichgeneralizes key properties of traditional single non-divisible object auctionsto the case where an arbitrarily divisible resource is to be shared. We haveshown that our auction rule, assuming an elastic-demand model of user pref-erences, constitutes a stable and e�cient allocation and pricing mechanismin a network context. Even though we are motivated by problems of band-width and bu�er space reservation in a communication network, the auctionwas formulated in a manner which is generic enough for use in a wide range

23

0 10 20 30 40 50 60 70 80 90 10080

100

120

140

160

180

200

Bid fee

num

ber

of b

ids

Figure 7: Number of bids to converge vs �

0 10 20 30 40 50 60 70 80 90 1001935

1940

1945

1950

1955

1960

1965

Bid fee

tota

l val

ue o

f allo

catio

ns

Figure 8:P

i �i(a�i ) { solid line, and maxa

Pi �i(ai) � 4(��)�1=2 { dashed

line, vs �.

24

of situations. In the sequel to this work, we show that the key results {namely incentive compatibility, equilibrium, and e�ciency { generalize to asetting where multiple networked resources are auctioned, with users bid-ding on arbitrary but �xed routes and topologies [29, 31]. In related work,we consider the case of stochastically arriving players bidding for advancereservations (i.e. resources for a given period of time) [29, 30].

An interesting direction of future work is learning strategies, and evolu-tionary behavior which can emerge from repeated inter-action between thesame players.

5 Acknowledgements

The authors would like to thank Wang Ke for his valuable help in the sim-ulations.

25

A Information-theoretic basis for the valuation

In general, valuations are simply assumed to be given as external factors.Indeed, the fundamental assumption in any market theory is that buyersknow what the goods are worth to them. The \elastic demand" or \di-minishing returns" nature of Assumption 1 is fully justi�ed from a purelyeconomic standpoint for virtually all resources in everyday life.

In the case of variable bandwidth, we can go even further by better quan-tifying what the goods are. For a user sending video, say, how much value islost when the channel capacity goes from 1.5 to 1.2 Mbps? Ultimately, thevalue lies not in the amount of raw bandwidth but in the information that issuccessfully sent. Our goal in this section is to give a brief description of howInformation Theory can be used for a bottom-up construction of bandwidthvaluations { based on the fundamental thing the user cares about which iscommunication of information { and that such valuations will generally beof the type in Assumption 1.

Any information source has a function D(:), such that when compressedto a rate R, the signal has a distortion of at least D(R) [1]. The distor-tion is the least possible expected \distance" between the original and com-pressed signals, where the minimization is over all possible coding/decodingschemes. In this context, we make the distance measure the monetary costof the error. This cost can be chosen, for example, to be proportional tosome common measures like the mean squared error, the Hamming distance(probability of error), the maximum error, etc., or heuristic measures basedon experiments with human perception. Given that modern source-codingtechniques can, given a distortion measure, achieve distortions close to thetheoretical lower bound[5], it is not unreasonable to use the rate-distortioncurve as an indication of the value of the bandwidth share.

Let Di(:) be the distortion-rate function of fXi(t)g, a stochastic processmodeling the source of information associated with user i. Xi is encoded asYi which has a rate of R bits per second.

Shannon's channel coding theorem[32] states that Yi can be receivedwithout errors if and only if the channel has a capacity C > R. In ourauction context, user i has capacity (bandwidth allocation) C = ai, andthus has to su�er a distortion of at least Di(ai). The value of the bandwidthis then

�i(ai) = �i �Di(ai); (18)

where �i is the value of the full information.

26

The relevant properties of the distortion-rate functions are:

� when the rate is greater than the entropy of the source, the distortionis zero, and

� for many common source models and cost functions, the distortion-rate function is convex, and has a continuous derivative.

It is easy to see that, with these properties, (18) satis�es Assumption 1.Example 1: Let fXg be a Bernoulli source, taking two values with

probabilities p and 1� p. Without loss of generality, let p 2 [0; 1=2]. In thiscase, since the source is i.i.d, one can de�ne the distortion on a per symbolbasis. Using a Hamming cost function

d(X; Y ) = 1fX 6=Y g;

i.e. assuming it costs one unit of money every time one bit is wrong, we havethe distortion D = Ed(X; Y ) = P (X 6= Y ). the rate-distortion function is

R(D) = [H(p)�H(D)]+ ;

where H(x) = �x log(x)�(1�x) log(1�x), and the distortion-rate functionis the inverse function. It can be easily seen that D(R) is strictly convex anddecreasing for 0 � R < H(p), and D(R) = 0 for R � H(p). The continuityof D0 on 0 � R < H(p) and R > H(p) is obvious. At the critical point(R = H(p); D = 0), we have

limR%H(p)

D0(R) = limD&0

1=R0(D)

= limD&0

1= log(D=1�D)

= 0

= limR&H(p)

D0(R):

Thus continuity of D0 holds throughout, and Assumption 1 is valid for thevaluation of the form (18) for this source { see Figure 9.

Example 2: Let fXg be a Gaussian source with Markovian time-

dependency, i.e a covariance matrix � =��2rji�jj

�i;j; r 2 [0; 1). Suppose we

use the squared error cost, i.e. it costs one unit of money for one unit of en-ergy in the error signal. Then, we have for low distortionsD � (1�r)=(1+r),R(D) = 1

2 log1�r2

D , or

D(R) = (1� r2)�22�2R;

27

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

Rate

Val

uatio

n

Figure 9: Distortion-rate based valuation for a Bernoulli p = 1=2 source

and Assumption 1 clearly holds for (18). In the i.i.d. case (r = 0), theformula holds for all R.

As the source models get more complex, it rapidly becomes impossibleto give closed-form expressions for either R(D) or D(R). Often parametricforms are available, and the functions can be evaluated numerically. Fortu-nately, the convexity property extends to a wide class of models, includingfor example auto-regressive sources, even when the generating sequence isnon-Gaussian (see [1] for a full treatment of R(D), including the abovecases).

It can happen, e.g. for some video source models, that the R-D curve,which gives the best (R,D) pairs achievable by any coder/decoder, is notconvex. But, for tractability, practical codecs are usually optimized on aconvex hull of the space of possible (R,D) pairs[26, 27]. Thus, even whenthe theoretical D(R) curve is not convex, the actual distortion achieved inreal-life systems almost always varies in a convex manner with the availablebandwidth.

28

B Simulation software and bidding algorithm

A prototype software agent based implementation of the auction game,called TREX, has been developed and extensively used since December 1995.Much of the intuition behind the mechanism design and the analysis in thiswork came from experiments done on this inter-active distributed auctiongame on the World Wide Web, using the Java programming language. Thegame can be played in real-time by any number of players from anywhereon the Internet [28].

Each user plays in the dynamic auction game using the following:

Algorithm 1 1 Let si = 0, and s�i = ;. Start an independentthread which receives updates of s�i.

2 Compute the truthful �-best-reply of Proposition 1, ti 2 Ti \S�i (s�i).

3 If ui(ti; s�i) > ui(si; s�i) + �, then send the bid si = ti.

4 Sleep for 1 second.

5 Go to 2.

No assumption is made on the order of the turns. Players join the gameat di�erent times, and depending on the execution context of the clientprogram, the sleep time of 1 second is more or less approximate. This, alongwith communication delays which make the times at which bids arrive atthe server and updates at the clients essentially random times, make thedistributed game completely asynchronous.

Algorithm 1 can be described as sel�sh and short-sighted. Sel�sh becauseit will submit a new bid if and only if it can improve it's own utility (by morethan the fee for the bid). Thus, the game can only converge7 to an �-Nashequilibrium. And short-sighted because it does not take the extensive formof the game into account, i.e. does not use strategies which may result in atemporary loss but a better utility in the long run.

References

[1] T. Berger. Rate Distortion Theory: A Mathematical Basis for DataCompression. Prentice-Hall, 1971.

7The fact that it does converge is established by Proposition 4 in Chapter 2 of [29].

29

[2] D. Clark. Internet cost allocation and pricing. In L. W. McKnight andJ. P. Bailey, editors, Internet Economics. MIT Press, 1997.

[3] E. H. Clarke. Multipart pricing of public goods. Public Choice, 8:17{33,1971.

[4] D. Fudenberg and J. Tirole. Game Theory. MIT Press, 1991.

[5] A. Gersho and R. M Gray. Vector Quantization and Signal Compres-sion. Kluwer Academic Publishers, 1992.

[6] J. Gong and S. Marble. Pricing common resources under stochasticdemand. preprint { Bellcore, April 1997.

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